Curvilinear 2|3D finite elements for the analysis of shells with arbitrary curvature and variable thickness

This paper presents a novel curvilinear finite element (FE) formulation for the static and modal analysis of shells with arbitrary curvature and variable thickness. In the proposed approach, high-order 2D shell models are defined in three curvilinear coordinates, exploiting the co- and contravariant components of physical quantities in the non-orthogonal reference frame. The Carrera Unified Formulation (CUF) is adopted for the definition of 2D shell models in which Lagrange functions are employed for the through-the-thickness approximation of displacements; then, merging CUF with finite element approximation of midsurface, new 3D-like FE elements are generated in which different orders of expansion can be adopted along the three curvilinear coordinates. For this reason, these elements are referred to as 2|3-D because they present both the computational efficiency of 2D models and the capability to model non-orthogonal geometries, such as shells with variable thickness, of 3D elements. Geometrical relations are derived within the classical differential geometry framework, and weak-form equilibrium equations are derived through the Principle of Virtual Displacements (PVD). The capabilities of the present finite elements are investigated by performing the static and modal analysis of various shell-like structures. The accuracy of the present approach in terms of three-dimensional stress states and natural frequencies is demonstrated by comparing the numerical results with solutions obtained by classical 3D finite elements using commercial software, highlighting the computational efficiency of the present elements. Finally, the proposed methodology is applied to analyze complex engineering applications.